The present invention relates generally to the generation of arbitrary waveforms, and more particularly to a method and apparatus for synthesizing and for utilizing such waveforms.
Various types of waveforms and waveform generators are used not just in technical fields, but also in numerous industrial and commercial applications. This is particularly true in electrical and electronic technologies, and perhaps even more importantly in optical technologies such as fiber optic data transmission. The needs are so demanding that more and more highly versatile mathematical techniques are required for generating a seemingly limitless variety of waveforms, and to handle the demands of technologies, such as communication and measurement, that are constantly increasing in speed.
Waveforms can be represented by mathematical functions, and ideally, the waveforms can then be realized or created by combining or superimposing certain groups of single-frequency components ranging from a frequency of zero to a frequency that is nearly infinite. In actuality, however, there are upper limits to the frequencies that can be utilized in real-world systems because of frequency response limitations in the equipment and the transmission lines. This means, as a practical matter, that frequency components at extremely high frequencies may not be available. Such upper frequency limitations then degrade the precision with which waveform generators can actually create the desired waveforms.
In theory, an ideal system could accurately generate virtually any waveform (an xe2x80x9carbitraryxe2x80x9d waveform) and could specify the mathematical function that defines the desired xe2x80x9carbitraryxe2x80x9d waveform. A simple example of such arbitrary function waveform generation shows, however, how difficult this can be in practice. xe2x80x9cSawtoothxe2x80x9d waves are very common, uncomplicated waveforms that are needed and are very useful in all sorts of electronic applications. Yet sawtooth waveforms are surprisingly difficult to generate, particularly at higher frequencies, such as used in cell phones, satellite communications, wireless internet access, and so forth.
The difficulty with sawtooth waveforms is caused by the sharp (xe2x80x9cpoint-likexe2x80x9d) transitions between the increasing and decreasing sides of the waveform. To keep these transitions sharp, very high-frequency capabilities are required. Otherwise, the transitions become xe2x80x9cbluntedxe2x80x9d. Since most electronic and optical equipment is xe2x80x9cband-limitedxe2x80x9d (i.e., cannot carry frequencies in the highest frequency bands), it is difficult in real-world systems to accurately propagate even a simple sawtooth voltage waveform. Similar considerations actually make it difficult even to accurately generate or create such a waveform in the first place (at higher frequencies). As can be appreciated, similar problems are presented with other waveforms that are more complicated.
The prior art presents many analytical approaches and proposes a number of solutions for these problems. Techniques are available for generating desired waveforms within a limited frequency bandwidth utilizing band-limited mathematical functions. However, generating such mathematical functions is not easy, both in the case of analog generation and at high frequencies. Accordingly, there continues to be a need for simpler, less complicated methods for generating function waveforms. Furthermore, in cases where distortion of the waveform occurs in a band-limited propagation medium, it is desirable to be able to correct this distortion.
Solutions to these problems have been long sought but prior developments have not taught or suggested any solutions and, thus, solutions to these problems have long eluded those skilled in the art.
The present invention provides a method for synthesizing an arbitrary waveform that approximates a specific waveform. Respective frequencies of component waveforms to be used to generate the arbitrary waveform are specified, the frequencies being less than the maximum frequency needed to synthesize the specific waveform. A least squares optimization of respective amplitudes and phases of the component waveforms is performed across at least one predetermined time interval. The component waveforms having the amplitudes and phases optimized by the least squares optimization are then summed to produce the arbitrary waveform. This method provides a simpler, more cost-effective means of generating an ideal waveform approximation at high frequencies.
Certain embodiments of the invention have other advantages in addition to or in place of those mentioned above. The advantages will become apparent to those skilled in the art from a reading of the following detailed description when taken with reference to the accompanying drawings.